Lagrangian Formalism#

The Lagrangian formalism of classical mechanics is a framework which can be used to predict how a system of point particles evolves over time. It is based on an empirically derived principle stating that for each set of generalized coordinates and their generalized velocities there is a special function such that the system evolves in such a way so as to either minimize or maximize a particular integral.

Definition: Lagrangian

Let \(q_1, \dotsc, q_s\) and \(\dot{q}_1, \dotsc, \dot{q}_s\) are any generalized coordinates and their generalized velocities for a system \(\mathcal{S}\) of point particles.

A Lagrangian of \(\mathcal{S}\) is function

\[ \mathcal{L}(q_1, \dotsc, q_s, \dot{q}_1, \dotsc, \dot{q}_s, t) \]

of \(q_1, \dotsc, q_s\), \(\dot{q}_1, \dotsc, \dot{q}_s\) and the time \(t\).

Definition: Action

Let \(t_1\) and \(t_2\) be two moments in time.

The action of a path \(\boldsymbol{q}: [t_1; t_2] \to \mathbb{R}^s\) in configuration space is defined via the following integral:

\[ S[\boldsymbol{q}] \overset{\text{def}}{=} \int_{t_1}^{t_2} \mathcal{L}(q_1(t), \dotsc, q_s(t), \dot{q}_1(t), \dotsc, \dot{q}_s(t), t) \mathop{\mathrm{d}t} \]

The Lagrangian formalism is based on the principle that for each set of generalized coordinates and their generalized velocities there is a Lagrangian, usually called the Lagrangian of the system, which can be used to accurately predict the evolution of the system through time.

Axiom: Principle of Stationary Action

For each set of generalized coordinates and their generalized velocities for a system of point particles there exists a Lagrangian \(\mathcal{L}\) such that, between any moments \(t_1\) and \(t_2\), the path of the system in configuration space is the one whose action is either the lowest or highest possible out of all paths.